# M\"obius-Frobenius maps on irreducible polynomials

## TL;DR

This paper explores the action of M"obius-Frobenius maps on irreducible polynomials over finite fields, characterizing fixed points and their counts, thus advancing understanding of polynomial symmetries under group actions.

## Contribution

It introduces a natural group action of the Projective Semilinear Group on irreducible polynomials and characterizes fixed points, providing new insights into polynomial symmetries in finite fields.

## Key findings

- Characterization of fixed points under the group action
- Enumeration of fixed irreducible polynomials
- Insights into symmetries of polynomials over finite fields

## Abstract

Let $n$ be a positive integer and let $\mathbb F_{q^n}$ be the finite field with $q^n$ elements, where $q$ is a power of a prime. This paper introduces a natural action of the Projective Semilinear Group $\text{P}\Gamma \text{L}(2, q^n)=\text{PGL}(2, q^n)\rtimes \text{Gal}(\mathbb{F}_{q^n}/\mathbb{F}_q)$ on the set of monic irreducible polynomials over the finite field $\mathbb{F}_{q^n}$. Our main results provide information on the characterization and number of fixed points.